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O-Level Additional Mathematics Tuition Singapore
Solving Exponential Equations
In solving exponential equations, it is common to use one or both of the following methods:
Method 1
Expressing both sides in the same base, then equating the indices.
Method 2
Substitution
Example
2^x = 32
2^x = 2^5
Therefore x = 5
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Quadratic Equation
ax^2 + bx + c = 0
Sum of the roots = – (b/a)
Product of roots = (c/a)
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Sketching a Quadratic Curve
Summary
1. Shape of the curve; a minimum curve or a maximum curve
When a>0 — a minimum curve
When a<0 — a maximum curve
2. Axes-intercept
Set y = 0 and find x
Set x = 0 and find y
3. Value of the turning point
4. End points if domain is given
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S3 – Teaching Indices and practice exam questions
S4 – Revising Binomial Expansions and practice exam questions
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Simultaneous Linear Equations
Learning Objective
Solve 2 simultaneous linear equations using elimination or substitution method
Non-Linear Equations
Learning Objective
Identify non-linear equations and use substitution method to solve the equations.
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Indices & Surds
4 Keys Areas
– Rules of indices and surds
– Solving exponential equations
– Simplifying surds
– Denominators containing surds ie rationalising denominator.
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Logarithms and Modulus Function
Objective
1. Know and use the laws of logarithms to simplify a logarithmic function or solve a logarithmic equation.
2. Know simple properties of common logarithmic and natural logarithmic functions.
3. Know and use the formula to change base.
4. Understand the properties of logarithmic functions and draw the graph.
5. Solve simple equations of modulus function lxl and sketch the graph of lf(x)l
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Trigometry
Sine Rule
a/Sin A = b/Sin B = c/Sicn C
Cosine Rule
a^2 = b^2 + c^2 – 2bcCosA
Area of triangle = (1/2)abSinC
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Study Notes – Equations and Inequilities
1. Simultaneous linear and non-linear equations can be solved by
(a) the algebraic method (substitution),
(b) the graphical method (finding the points of intersection).
2. Given that x1 and x2 are the roots of the equation
ax^2 + bx + c = 0 where a is not 0,
(a) sum of roots = x1 + x2 = — —b/c
(b) product of roots = (x1)(x2) = c/a
3. A quadratic equation with roots x1 and x2 is given by
(x — x1)(x — x2) = 0 or
x^2- (x1 + x2)x + (x1)(x2) = 0.
4. To solve a problem on intersection of a line and a curve,
(a) equate the two given equations to form a quadratic equation in x or y,
(b) use the relationship between the discriminant and the number of points of intersection to solve the given problem.
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Conditions for quadratic equation to have
(1) Two real and equal roots (b2 — 4ac = 0)
(2) Two real and distinct roots (b2 — 4ac > 0)
(3) Two real roots (b2 — 4ac 0)
(4) No real roots (b2 — 4ac < 0)
Quadratic Functions and Inequalities.
1. Find the range of values of m for which the equation (m + 2)x2 — 2mx + 1 = 0 has two complex roots.
2. Given that y = 16 — px — x2 = 25 — (q + x)2, for all real values of x, where p and q are both positive
(i) calculate the values ofp and q,
(ii) state the maximum value of y and the value of x at which it occurs
(iii) find the range of values of x for which y is positive.
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PROBLEMS INVOLVING INTERSECTION OF LINE AND CURVE
These problems can be solved using the concepts we have learned in this section, as follows:
Equation of line: nix + my = p
Equation of curve: f(x, y) = q whose f (x, y) = any function of x and y
Step (1) Solve the 2 equations simultaneously. either by: elimination method or by: substitution method.
Step (2) The result of step (1) is (after simplification) a quadratic equation (ax^2 + bx + c = 0) and we now apply the following rules:
If line intersects curve at two points
=> quadratic equation (ax^2 + bx + c = 0) has two distinct roots
=> b^2 — 4ac > 0
Or
If line intersects curve at 1 point only (or line is tangent to the curve)
=> quadratic equation (ax^2 + bx + c = 0) has one repeated root
=> b^2 — 4ac = 0
Or
If line does not cut or intersects the curve quadratic equation
(ax^2 + bx + c= 0) has no real roots (so b^2 — 4ac < 0)
Step (3) After solving for one variable [from step (2)] substitute to find the value of other variable. We must have two, one or no points of intersection.
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Remainder Theorem
When a function f (x) — usually a polynomial function, y = a0 + a +
a2x2 + a3x3 +……….
where a = constant is divided by a linear factor (x — k) the remainder upon Long Division is the value f (k).
Factor Theorem
This is a special case of the Remainder Theorem. When a function f (x) is divided by a liner factor (x — k) and the remainder = 0, then (x — k) is a factor off (x).
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Simultaneous equations in 2 unknowns
Solving Simultaneous equations in 2 unknowns x, y can be done in three ways
• Substitution
• Elimination
• Matrix method (only if both equations are linear)
A. Use of substitution method
Solve for x, y:
x^2- 2xy -y^2= 0
x – 2y = 4
Answers: y = -1, x = 2 and y =- 3 , x = 4
B. Use of elimination method
Solve for x, y:
2x – 3y = 7
-4x + 19y = -1
Answer: x = 5, y = 1
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TRIGONOMETRY
1. GENERAL ANGLES
• General angles
• Basic angles
General angles
In a cartesian plane, for an angle A measured from the positive x-axis, A is a positive angle if it is measured in an anti-clockwise direction. A is a negative angle if it is measured in a clockwise direction.
Basic angles
The angle, a , which is the positive acute angle (an angle between 0° and 90°)made by the arm OP and the x-axis, is known as the basic angle, associated angle or the reference angle.
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